ON THE SUM OF THE TWISTED EULER FUNCTION

Abstract

We study an asymptotic formula for the sum of values of the Euler φ-function twisted by a real Dirichlet character. The error term is split into the arithmetic and the analytic part. The former is studied with minimal use of analytic tools in contrast to the latter, where the analysis depends heavily on the distribution of the non-trivial zeros of the corresponding Dirichlet L-function. The results of the present paper are an extension of a recent work by the authors, where the case of the classical Euler φ-function has been studied. The present, more general situation invites new technical difficulties. Not all of them can be successfully overcome. For instance, satisfactory omega results for the analytic part are proved in the case of an even Dirichlet character only. Nevertheless, a method providing good omega estimates for the arithmetic part as well as for the complete error term is developed. Moreover, it is noted that the Riemann Hypothesis for the involved Dirichlet L-function is equivalent to a sufficiently sharp estimation of the analytic part. This shows in particular that the arithmetic part can be much larger than the corresponding analytic part.